The average height of American males (over 16) is known to be 69.2”. This inform

Question

The average height of American males (over 16) is known to be 69.2”. This information is from a previous study. Answer the following questions using the Height (HT) column as your sample.

   (a) You believe that the average height of males is actually less than 69.2”. Test this claim and assume that = 3.12”.

   (b) You believe that the average height of males is actually less than 69.2”. Test this claim and assume that is unknown.

   (c) Test the claim that the standard deviation of male heights is less than 3.12”, using the value of from part (a) as your known population s.d.

Heights

70.8 66.2 71.7 68.7 67.6 69.2 66.5 67.2 68.3 65.6 63.0 68.3 73.1 67.6 68.0 71.0 61.3 76.2 66.3 69.7 65.4 70.0 62.9 68.5 68.3 69.4 69.2 68.0 71.9 66.1 72.4 73.0 68.0 68.7 70.3 63.7 71.1 65.6 68.3 66.3

Explanation / Answer

(a)

Data:    

n = 40   

= 69.2   

s = 3.12   

x-bar = 68.335   

Hypotheses:    

Ho: 69.2   

Ha: < 69.2   

Decision Rule:    

= 0.05   

Critical z- score = -1.644853627   

Reject Ho if z < -1.644853627   

Test Statistic:    

SE = s/n = 3.12/40 = 0.493315315

z = (x-bar - )/SE = (68.335 - 69.2)/0.493315314986267 = -1.753442421

p- value = 0.039763047   

Decision (in terms of the hypotheses):    

Since -1.753442421 < -1.644853627 we reject Ho

Conclusion (in terms of the problem):    

There is sufficient evidence that < 69.2

(b)

Data:    

n = 40   

= 69.2   

s = 3.02   

x-bar = 68.335   

Hypotheses:    

Ho: 69.2   

Ha: < 69.2   

Decision Rule:    

= 0.05   

Degrees of freedom = 40 - 1 = 39

Critical t- score = -1.684875122   

Reject Ho if t < -1.684875122   

Test Statistic:    

SE = s/n = 3.02/40 = 0.477503927

t = (x-bar - )/SE = (68.335 - 69.2)/0.477503926685425 = -1.811503428

p- value = 0.03888469   

Decision (in terms of the hypotheses):    

Since -1.811503428 < -1.684875122 we reject Ho and accept Ha

Conclusion (in terms of the problem):    

There is sufficient evidence that < 69.2

(c)

Data:     

n = 40    

= 3.12    

s = 3.02    

Hypotheses:     

Ho: 3.12    

Ha: < 3.12    

Decision Rule:     

Degrees of freedom = n - 1 =    40 - 1 = 39

= 0.05    

Critical value =   25.69539078   

Reject Ho if the test 2 value <    25.69539078

Test Statistic:     

2 = (n - 1) s^2 / ^2 =   (40 - 1) * 3.02^2 / 3.12^2 = 36.5400641

p- value = 0.41736612    

Decision (in terms of the hypotheses):     

Since 36.5400641 > 25.69539078 we do not reject Ho

Conclusion (in terms of the problem):     

There is no sufficient evidence that the population standard deviation is less than 3.12.

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